NCSU Differential Equations/Nonlinear Analysis Seminar Schedule Spring 2021

Wednesday, February 24, 15:00-16:00

Zoom meeting: Link
Speaker: Andrew Papanicolaou, North Carolina State University, USA
Title: Consistent Inter-Model Specification for Time-Homogeneous SPX Stochastic Volatility and VIX Market Models
Abstract: This work explores the recovery stochastic volatility models (SVMs) from market models for the VIX futures term structure. Market models have more flexibility for fitting of curves than do SVMs, and therefore they are better-suited for pricing VIX futures and derivatives. But the VIX itself is a derivative of the S&P500 (SPX) and it is common practice to price SPX derivatives using an SVM. Hence, a consistent model for both SPX and VIX derivatives would be one where the SVM is obtained by inverting the market model. Analysis will show that some conditions need to be met in order for there to not be any inter-model arbitrage or mis-priced derivatives. Given these conditions the inverse problem can be solved.
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Wednesday, March 03, 15:00-16:00

Zoom meeting: Link
Speaker: Marta Lewicka, University of Pittsburgh, USA
Title: Expansions of averaging operators and applications
Abstract: In my talk, I will explain the approach of finding solutions to nonlinear PDEs via tug-of-war games. I will focus on the context of p-Laplacian and the non-local geometric p-Laplacian.
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Wednesday, March 10, 15:00-16:00

Zoom meeting: Link
Speaker: Hédy Attouch, Université Montpellier II, France
Title: Acceleration of first-order optimization algorithms via inertial dynamics with Hessian driven damping
Abstract: In a Hilbert space, for convex optimization, we report on recent advances regarding the acceleration of first-order algorithms. We rely on inertial dynamics with damping driven by the Hessian, and the link between continuous dynamic systems and algorithms obtained by temporal discretization. We first review the classical results, from Polyak’s heavy ball with friction method to Nesterov’s accelerated gradient method. Then we introduce the damping driven by the Hessian which intervenes in the dynamic in the form D^2f(x(t))x'(t). By treating this term as the time derivative of Df(x(t)), this gives, in discretized form, first-order algorithms.As a fundamental property, this geometric damping makes it possible to attenuate the oscillations. In addition to the fast convergence of the values, the algorithms thus obtained show a rapid convergence towards zero of the gradients. The introduction of time scale factors further accelerates these algorithms. On the basis of a regularization technique using the Moreau envelope, we extend the method to non-smooth convex functions with extended real values. Numerical results for structured optimization problems support our theoretical findings. Finally, we evoke recent developement concerning the extension of these results to the case of monotone structured inclusions, inertial ADMM algorithms, dry friction, inexact/stochastic case, Tikhonov regularization, non-convex tame optimization, closed-loop adaptive damping, thus showing the versatility of the method.
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Wednesday, March 17, 15:00-16:00

Zoom meeting: Link
Speaker: Alexander Reznikov, Florida State University
Title: Discrete minimization problems on non-rectifiable sets
Abstract: We survey the known results concerning the minimal Riesz energy on sufficiently smooth sets, and present some new results on fractal sets, which are the key examples of non-rectifiable sets. In particular, we will talk about the connection of these problems to ergodic theory and probability theory: the key step in our proofs is applying the so-called “renewal theorem”.
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Wednesday, March 31, 15:00-16:00

Zoom meeting: Link
Speaker: Roberto Cominetti, Universidad Adolfo Ibáñez, Chile
Title: Convergence rates for Krasnoselskii-Mann fixed-point iterations
Abstract: A popular method to approximate a fixed point of a non-expansive map is C is the Krasnoselskii-Mann iteration. This covers a wide range of iterative methods in convex minimization, equilibria, and beyond. In the Euclidean setting, a flexible method to obtain convergence rates for this iteration is the PEP methodology introduced by Drori and Teboulle (2012), which is based on semi-definite programming. When the underlying norm is no longer Hilbert, PEP can be substituted by an approach based on recursive estimates obtained by using optimal transport. This approach can be traced back to early work by Baillon and Bruck (1992, 1996). In this talk we describe this optimal transport technique, and we survey some recent progress that settles two conjectures by Baillon and Bruck, and yield the following tight metric estimate for the fixed-point residuals.

The recursive estimates exhibit a very rich structure and induce a very peculiar metric over the integers. The analysis exploits an unexpected connection with discrete probability and combinatorics, related to the Gambler’s ruin for sums of non homogeneous Bernoulli trials. If time allows, we will briefly discuss the extension to inexact iterations, and a connection to Markov chains with rewards.

The talk will be based on joint work with Mario Bravo, Matias Pavez-Signé, José Soto, and José Vaisman.
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Wednesday, April 07, 15:00-16:00

Zoom meeting: Link
Speaker: Tyrus Berry, George Mason University
Title: A Manifold Learning Approach to Boundary Value Problems
Abstract: Mesh-free methods for boundary value problems (BVPs) can be convenient on manifolds where generating a mesh may be difficult or when the manifold is not known explicitly but is determined by data.  Moreover, BVPs are important in machine learning since they provide a rigorous method of regularization for many regression problems.  In this talk we introduce the tools required to solve basic BVPs using only points sampled on an unknown manifold embedded in Euclidean space.  The key advance, discovered by Ryan Vaughn, is that the Diffusion Maps algorithm from machine learning is a consistent estimator of the Dirichlet energy (or weak-sense Laplacian) for manifolds with boundary.  Explaining this surprising and challenging result will form the core of the presentation and provides many insights valuable to kernel-based machine learning methods.  This result is also the key to developing the other components necessary for solving BVPs, including an estimator for the distance-to-boundary function and a boundary integral estimator.  Finally, these tools are combined to solve the weak formulation of standard BVPs on manifolds.

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Wednesday, April 14, 15:00-16:00

Zoom meeting: Link
Speaker: Alessio Porretta, Università di Roma Tor Vergata
Title: Long time behavior in mean field game systems
Abstract: Mean field game PDE systems were introduced by J-M. Lasry and P.-L. Lions to describe Nash equilibria in multi-agents dynamic optimization. In the simplest model, those are forward-backward systems coupling Hamilton-Jacobi with Fokker-Planck equations. In this talk I will discuss the long time behavior of second order systems in the periodic case under suitable stability conditions. I will go through the main features that appear in the study of the long time limit: the ergodic behavior, the effects of the forward-backward structure, the exponential turnpike property of the underlying control problem and the link with the vanishing discount limit in infinite horizon problems.
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Wednesday, April 21, 15:00-16:00

Zoom meeting: Link
Speaker:  Barbara Kaltenbacher, University of Klagenfurt
Title: Some Asymptotics of Equations in Nonlinear Acoustics
Abstract: High intensity (focused) ultrasound HIFU is used in numerous medical and industrial applications ranging from lithotripsy and thermotherapy via ultrasound cleaning and welding to sonochemistry. The relatively high amplitudes arising in these applications necessitate modeling of sound propagation via nonlinear wave equations and in this talk we will first of all dwell on this modeling aspect. Then in the main part of this lecture will deal with limiting cases of certain parameters, that are both of physical interest and mathematically challenging. The latter is due to the fact that these limits are singular in the sense that they change the qualitative behaviour of solutions. On a technical level, they require uniform bounds and therefore alternative energy estimates. We start with the classical Westervelt and Kuznetsov models and study the limit as the diffusivity of sound tends to zero – the parameter of a viscous damping term whose omission leads to a loss of regularity and global well-posedness as well as exponential decay. Secondly, we consider the Jordan-Moore-Gibson-Thompson equation, a third order in time wave equation that avoids the infinite signal speed paradox of classical second order in time strongly damped models of nonlinear acoustics, such as the already mentioned Westervelt and the Kuznetsov equation. We study the limit as the parameter of the third order time derivative that plays the role of a relaxation time tends to zero, which again leads to the classical Kuznetsov and Westervelt models. Making such a transition from third order to second order in time equations clearly necessitates compatibility conditions on the initial data. Finally, we provide a result on another higher order model in nonlinear acoustics, the Blackstock-Crighton-Brunnhuber-Jordan equation for vanishing thermal conductivity. 

This is joint work with Vanja Nikolic, Radboud University, and Mechthild Thalhammer, University of Innsbruck.
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Wednesday, April 28, 15:00-16:00

Zoom meeting: Link
Speaker: Jérôme Bolte, Université Toulouse 1 Capitole
Title: A Bestiary of Counterexamples in Smooth Convex Optimization
Abstract: Counterexamples to some old-standing optimization problems in the smooth convex coercive setting are provided. Block-coordinate, steepest descent with exact search or Bregman descent methods do not generally converge. Other failures of various desirable features are established: directional convergence of Cauchy’s gradient curves, convergence of Newton’s flow, finite length of Tikhonov path, convergence of central paths, or smooth Kurdyka- Lojasiewicz inequality. All examples are planar. These examples rely on a new convex interpolation result: given a decreasing sequence of positively curved C^k smooth convex compact sets in the plane, we provide a level set interpolation of a C^k smooth convex function where k ≥ 2 is arbitrary. If the intersection is reduced to one point our interpolant has positive definite Hessian, otherwise it is positive definite out of the solution set. (Joint work with E. Pauwels (Université Toulouse 1)
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